Recall from last lecture we studied

• Random sampling: where we select the sample from the population in an unbiased fashion
• Bias: A sample is biased if some observational units have a higher probability of being included in the sample.

Keywords: random and probability

• In short: Probability is the study of randomness.
• It's roots lie in one historical constant.

In 79 AD a volcanic eruption in the Roman city of Pompeii covered the entire city in hot ash, perfectly preserving may items:

including…

There are two way to approaches to studying probability:

Mathematically (MATH 310)	Simulations

• The mathematical approach to studying probability requires more mathematical background: set theory, discrete math, calculus, and later on measure theory.
• The simulation approach does not. Instead we requires a computer's random number generator to generate simulations. Why?

Doing this repeatedly by hand is tiring:

The "random simulation" in question is not limited to coin flips.

For any random simulation, think to this: POWERBALL

We run different simulations by tinkering with the Powerball:

• Balls:
  • How many balls do you have?
  • What are written on the balls?
  • Do the balls have equal probability of being picked?
• Drawing balls:
  • How are you drawing the balls
  • How many balls do you pick?
  • What are you recording from each ball?
  • What to do with a ball after you've picked/recorded it?
• Number of lotteries:
  • How many times do you repeat the lottery?

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## 
## Flipping 1 coin [ Prob(Heads) = 0.5 ] ...
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## T
## 
## Number of Heads: 0 [Proportion Heads: 0]

## 
## Flipping 10 coins [ Prob(Heads) = 0.5 ] ...
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## H H T T H H H H T T
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## Number of Heads: 6 [Proportion Heads: 0.6]

## # A tibble: 10 × 4
##        n heads tails  prop
##    <dbl> <dbl> <dbl> <dbl>
## 1     10     6     4   0.6
## 2     10     7     3   0.7
## 3     10     4     6   0.4
## 4     10     6     4   0.6
## 5     10     5     5   0.5
## 6     10     4     6   0.4
## 7     10     3     7   0.3
## 8     10     4     6   0.4
## 9     10     6     4   0.6
## 10    10     5     5   0.5

## # A tibble: 5 × 2
##   heads count
##   <dbl> <int>
## 1     3     1
## 2     4     3
## 3     5     2
## 4     6     3
## 5     7     1

##   n heads tails prop
## 1 1     0     1    0
## 2 1     0     1    0
## 3 1     1     0    1

##   V1 V2 V3 V4 V5 V6
## 1  2  1  3  1  2  2
## 2  1  2  1  2  2  1
## 3  6  3  1  6  6  5

##   V1 V2 V3 V4 V5 V6
## 1  5  4  2  1  6  3
## 2  3  6  2  4  1  5
## 3  2  3  4  6  1  5